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Full text of "Treatise On Analysis Vol-Ii"

258      XIV   INTEGRATION IN LOCALLY COMPACT GROUPS

If G is either compact or discrete, then we have mod(w) = 1 for every
automorphism u of G. For it is clear that w(G) = G and u({e}) = {e}, and we
may apply (14.3.6.2) with A = G and A = {e}, respectively.

We deduce also from (14.3.6,2) that if u and v are two automorphisms of
G, then

(14.3.8)                         mod(w ° v) = mod(w) • mod(v).

(14.3.9)    Ifuis any automorphism of the vector space R", then

mod(w) = |det u\.

Let U = (%ij) be the matrix of the automorphism u with respect to the
canonical basis of R". Let Etj denote the n x n matrix which has the element in
the (ij) place equal to 1 and all other elements equal to 0. If / ^j and 1 e R,
put

Then we have the following lemma:

(14.3.9.1)    Every invertible n x n matrix U is a product of matrices of the
form Btj(X) and a matrix of the form In + (a~ l)Enn .

Consider invertible matrices of the form

/I   0   ••• 0   1   •••
	0   {lfB_fc
 0      £2,n-h
	£i»       \

0   0   ••-
	1    £n^h-lin-.h ' • •
	«.-*-!..

A      A
	A       e
	i:                       i

I

where /z^0;if/z = 7t-l, then .A" is an arbitrary invertible matrix. The proof
is by induction on h. The matrix BU(X)X is obtained by adding A times the
jth row to the fth row of X. If h = 0 we must have %nn ^ 0; hence if we mul-
tiply on the left successively by the matrices Bin(-%~nl£in) for 1 g i <£ n - 1,
we shall obtain the matrix /„ + (£nn - \)Enn, which proves the lemma in the
case h = 0. Now suppose that the lemma has been proved for A = 0, 1, ...,
k-\<n, and consider the case h = k. From the expansion for det(JT)
we see Ann 7.4 that there must exist an element £i>n-k ^ 0 for some / such
that n-k-^i^n. Premultiplying X by Bn((\ - £;> '„ -*)C«-*) for some index
j ^ i such that n-k^j^n, we see that we may assume that £/,„-* = 1-
Multiplying successively by the matrices ^r/(-^r)/J_fc) for r ^j9 we end uposed